A computational model coupling mechanics and electrophysiology in spinal cord injury

Antoine Jerusalem • Julián A. Garcia-Grajales • Angel Merchán-Pérez • José M. Peña

Abstract Traumatic brain injury and spinal cord injury have recently been put under the spotlight as major causes of death and disability in the developed world. Despite the important ongoing experimental and modeling campaigns aimed at understanding the mechanics of tissue and cell damage typically observed in such events, the differenti­ated roles of strain, stress and their corresponding loading rates on the damage level itself remain unclear. More specif­ically, the direct relations between brain and spinal cord tis­sue or cell damage, and electrophysiological functions are still to be unraveled. Whereas mechanical modeling efforts are focusing mainly on stress distribution and mechanistic-based damage criteria, simulated function-based damage cri­teria are still missing. Here, we propose a new multiscale model of myelinated axon associating electrophysiological impairment to structural damage as a function of strain

and strain rate. This multiscale approach provides a new framework for damage evaluation directly relating neuron mechanics and electrophysiological properties, thus provid­ing a link between mechanical trauma and subsequent func­tional deficits.

Keywords Computational model • Axon • Electrophysiology • Mechanics • Spinal cord injury

1 Introduction

Neuropathological and neurological traumatic brain injuries (TBIs) and spinal cord injuries (SCIs) ranging from simple edema or axonal injury to cognitive deficits or post-traumatic stress disorders have already been widely referenced for both animal models and humans (Hicks et al. 2010; Gold­stein et al. 2012). At the cellular level, these injuries have been more specifically linked to cytoskeleton alterations, neurotransmitter deficits, disruption of calcium homeostasis, mitochondrial disturbances, loss in membrane permeability, demyelination and even altered gene expression (Cernak and Noble-Haeusslein 2010; Ouyang et al. 2010; Alford et al. 2011; Peter and Mofrad 2012). At the functional level, recent efforts by Shi and coworkers have quantified the injury level by measuring the membrane integrity and compound action potential (CAP) propagation under ex vivo electrical stim­uli of guinea pig spinal cord white matter after tensile (Shi and Whitebone 2006), compressive (Ouyang et al. 2008) and blast loading (Connell et al. 2011).

Computational biomechanics simulations making use of finite element schemes have recently allowed for the identifi­cation of stress extrema and/or patterns at the tissue (Moore et al. 2009; Nyein et al. 2010; Cloots 2011; Cloots et al. 2013; Gupta and Przekwas 2013) and cell scales (Jerusalem

and Dao 2012) during TBI events (see these references for a complete literature review). Conversely, recent work building on the observation of "leaky" voltage-gated sodium ion chan­nel after trauma (Wang et al. 2009) has proposed a model of the resulting hyperpolarization-(left-)shifts of the ion chan­nel current (Boucher et al. 2012). This model successfully reproduces the effect of the trauma-induced blebbing1 of a membrane patch on the electrophysiological properties of this patch. Building on the same left-shift mechanism, a sim­ilar study was proposed by Volman and Ng (2013), indicat­ing a marked difference in the behavior of potentials in sub­threshold (<14mV of potential shift) and supra-threshold (>14mV) traumas. Other approaches focusing on demyeli-nation (a mark of multiple sclerosis) and its geometrical effect on the additional exposition of the membrane to the surround­ing medium date back to the modeling work of Waxman and Brill (1978) which has been recently extended by Babbs and Shi (2013) to explore the effect of drug treatment on the conduction in the damaged region. Despite this grow­ing interest in linking trauma and electrophysiological alter­ations, the intrinsic relation between mechanico-geometrical (stress, strain and strain rates) and electrophysiological alter­ations (potentials and currents) was left unmodeled in all these models, and a fully multiscale model of the funda­mental link between the electrophysiological functions of the neuron—more specifically the propagation of action poten­tials (APs)—and its mechanical integrity is still lacking.

To this end, we have followed a three-step procedure: (i) we first developed a mechanical framework accounting for the multiscale deformation of the axonal components within the tissue and the resulting structural damage; (ii) as a sec­ond step, we developed a coupling model composed of two (geometrical and damage-based) components aimed at link­ing this mechanical framework to electrophysiological model parameters; and (iii) a one-dimension representative axon was discretized, in which the nodes of Ranvier (NRs) and the internodal regions (IRs) have been modeled using the Hodgkin-Huxley model and Cable Theory (Hodgkin and Huxley 1952; Koch 1999), respectively. In this approach, IRs are modeled as passive cables due to the multiple isolating myelin sheaths covering the axon membrane, while voltage-gated ion channels responsible for the generation and propa­gation of the APs are concentrated at the NRs (Koch 1999). The approach is multiscale by nature, defining mechanical properties at the axon level, while linking the electrophys­iological properties of the corresponding membrane in the NRs or IRs to the deformation and damage of the overall axon. As such, the intrinsically heterogeneous microscop-

Axonal blebbing—or beading—consists of a series of undulating swellings at the NRs (Ochs et al. 1994; Markvin et al. 1999) lead­ing to a loss of adhesion of the lipid bilayer to the underlying cortical axoplasm (Boucher et al. 2012).

ical mechanical and electrophysiological properties of the axonal components are homogenized at the mesoscale. Such approach, already widely used in the mechanical descrip­tion of many types of materials [e.g., composite materials (Wu et al. 2013)] benefits from a lower computational cost and thus allow for simulations of coupled multiscale phe­nomena currently unavailable to mechanical models defined solely at the microscale.

Following calibration and validation against currently available experimental data, the proposed three-component model was finally shown to successfully predict the elec­trophysiological alterations ensuing a damaging mechanical event, while providing a new modeling tool able to extract data otherwise inaccessible through experimental means.

2 Materials and methods

The objective of coupling two mechanical and electrophysio­logical models into a single integrated model is to transfer the effects derived from an injuring mechanical cause into elec­trophysiological alterations affecting AP propagation, and ultimately cell behavior. The resulting model is presented in Fig. 1, where the mechanical, coupling (geometrical and damage-based) and electrophysiological models have been tagged as 1,2 and 3, respectively. In the following, the build­ing blocks of the overall model are described, followed by the finite difference scheme used to simulate the AP propagation impairment.

2.1 Mechanical model

Because of the complex association between the outer and inner faces of the axonal membrane (myelinated in the IRs) and its surrounding structures (extracellular matrix and cortical axoplasm), the microscopic fluid-like viscoelastic mechanical behavior of the membrane can be thought of as a fraction of the macroscopic mechanical behavior of the over­all tissue. As such, and in view of the principal role of the membrane in the electrophysiology of the cell, a mechani­cal model focused around the membrane should be the first logical step. However, a much larger body of experimental data is available for the full axon mechanical and growth properties. Additionally, the intrinsic intertwined mechani­cal dependencies between membrane, cortical axoplasm and major cytoskeletal components such as microtubules natu­rally imply a quasi-one-to-one stretch and growth behavior between all these components (Garcia et al. 2012). As a con­sequence, the passive and active behavior of the full axon is represented here by a Maxwell model (i.e., an elastic spring and a damper in series) and coupled to another viscous com­ponent representing the surrounding tissue, see Fig. 1-1. Note that other authors have also proposed an additional retracting

Internodal region (IR) Node of Ranvier (NR) •

Axon Axial strain 8a

©

Tit H ] -

Extracellular matrix viscosity

IR £mi

V Em - > •

t £n%a

NR

rB(Em)

Cmy(Sm) 1

Cm (6m)

rm(£mj

rmy(6mj

gNa(Sm)

gK(em)

gi.(£m)

X V

Cytoplasm

Myelin Layers Extracellular Matrix

-^h-

Extracellular Matrix

r ' IR

I'm ^ ^ ^ 1 Cm

rimyrmy^^^ I

Vrest ~ ~ ^ 1 1 Cmy/rini

r. r. N R

L r I

T EL T EK T EN,

I

0

Mechanical model Coupling Electrical model

Fig. 1 Electrophysiological-mechanical framework: 1 mechanical model, 2 coupling model (geometrical and damage-based), 3 electrophysiolog­ical model

mechanism (Dennerll et al. 1989). We have decided to omit it in our model, as such effects are a priori not experimentally present at the rates under study.

This model implies the consideration of three parame­ters :E,r¡f and r¡f, respectively, the microscopic axial axonal stiffness (i.e., the spring in the Maxwell model), the tissue active molecular reorganization viscosity (i.e., the damper in the Maxwell model), and the viscous molecular cross-linking between the axon and its surrounding structures (i.e., the damper in parallel with the Maxwell model). More specif­ically, the viscous component of newtonian viscosity r]f rep­resents membrane turnover, (de)polymerization (e.g., micro­tubules) and (de)lipidation (e.g., the membrane and myelin sheaths), as well as any other active molecular reorganization driven by mechanical stretch or during relaxation (Dennerll et al. 1989). The superscript symbol "±" of r]f and r¡f indi­cate that this value is a priori different under applied tension and during macroscopic stress-free relaxation. The micro­scopic axial axonal strain sustained by the molecular bonds of the axon (and thus membrane) is defined by emA and the macroscopic overall axial strain by ea.

A microscopic damage contribution with a corresponding strain eDia is finally considered so that the axonal stress

£D,a (71 =

= 0,

(70 + keva,

if 0\ < (70

otherwise £6D,a (2)

where ero is the damage stress threshold of the axon, and where k is a model parameter representative of the dam­age evolution. This coupling between reversible-elastic and irreversible-plastic behaviors is justified by the observation that a relaxation of the membrane following large plas­tic deformation exhibits an elastic-reversible behavior upon reloading (Wang et al. 2009). Again, the deformation and failure of all axonal components are assumed to occur simul­taneously.

Gathering Eqs. (1), (2), and the two newtonian compo­nents, the system of equation characterizing the full model reads:

(7i = £ ( £ m , a - £D,a)

(7i = ?jf ( ¿ a - ¿ m , a )

02 = r¡2 ¿ a

a = o\ + (72

((7i - (70 - ££D,a)¿D,a

(3)

0 and (7i < (7Q + ke^ •

(7i = E(eB £D,a) (1)

is only representative of the unbroken bonds. Inspired by metal plasticity yield evolution rules, an evolution equation of this irreversible damage strain is assumed to verify the following rule:

It should be emphasized that the proposed model is the simplest mechanical model accounting for the experimen­tally observed loading and gradual unloading of the axon (along with the damage component). As a consequence, the elasticity of the surrounding extracellular matrix is neglected as a first approximation, thus avoiding an increase in the size of the parameter space.

Additionally, despite large values of the macroscopic strain in the following simulations, the microscopic strain is less than 25% in the vast majority of the cases of this work, with larger values for the upcoming fast-severe cases and the fast-moderate case at t = 0 min (where the potential is anyway reduced to 0). As a consequence, the small strain assumption was considered acceptable as a first approxima­tion (see also Sect. 4 below).

Finally, the equations are presented in a quasi-static frame­work, thus avoiding the need for additional dynamic terms. This choice is motivated by the observation that the elec­trophysiological measurements corresponding to the loading conditions used in the following do not exhibit any dynamic oscillations (Shi and Whitebone 2006), indicating that mate­rial deformation is not driven by its inertia, but by a slower viscous behavior.

2.2 Geometrical coupling model

The geometrical coupling model, see Fig. 1-2, first includes a strain model providing the NR and IR specific axial micro­scopic axonal strains, respectively, ej^a and e^R

a, based on their "average" counterpart emA (i.e., the output of the mechanical model). As described in detail in Supplementary Material A, for a given structural composition of the white matter and a given average microscopic axial strain em^, we postulate the existence of a parameter v e [0, 1] such that the microscopic axial strains of the NRs and IRs are given by

.NR

„IR

X 1 (4)

1 'm,a

where x is the density of NRs per axonal unit length. Note that the three special cases v = 0, v = 1 and v = x cor­respond, respectively, to: no strain in the NRs (the deforma­tion is fully accommodated by the IRs), no strain in the IR (the deformation is fully accommodated by the NRs), and fully homogeneous strain in the whole axon. For simplic­ity, the notations will subsequently drop the "IR" and "NR" superscripts, and, unless indicated otherwise, all subsequent formulae apply to both regions (NR or IR) individually.

The second step of the coupling model links the micro­scopic surface membrane strain em to the microscopic axial strain emA by noting that, assuming axon incompressibility,

1 yr (5)

Under the same assumption, the diameter of an axon stretched macroscopically by ea is given by

d do

yr (6)

The membrane thickness is additionally assumed to be con­stant based on the assumption that the "height" of the lipid bilayer remains the same during loading: h = ho- In the previous equations, the subscripts "o" in ho and do refer to their respective resting state values (no strain). The «my

myelin layers in the IRs follow the same scaling laws for their own individual resting thickness /zmy,o and diameter dl

my0 = d0 + 2h0 + 2(i - l)/imy,o, where i e [1, nmy] is the number of the layer.

As a final step, these relations correlate the microscopic mechanical viscoelastic stretch of the axonal regions IR and NR (characterized by e^R

a and e|^a for the NRs and IRs, respectively) to their membrane surface-dependent electrical properties (characterized by the set of capacitances cm, cmy, resistances ra, rm and rmy, and the leak, voltage-gated sodium and potassium channels #L, gNa and £K); see Fig. 1-2 and Sect. 2.5 below. It is noticeable that these electrophysiolog­ical alterations are thus related to the change in membrane surface area, i.e., geometrically driven.

2.3 Electrophysiological model

The myelinated IR segments of the axon are simulated according to the Cable Theory as an electrical circuit involv­ing the resting transmembrane potential (Vrest), the axial resistivity of the axonal cytoplasm (pa), the transmembrane resistivity (pm), the resistivities of the myelin sheaths (pmy), and the cell membrane and myelin sheaths equivalent elec­tric constants (Cm and Cmy). Simulation of the NRs by the Hodgkin-Huxley model are based on the same parameters (except those pertaining to myelin and the membrane resis­tivity) plus the variable conductivities (GN 3 and GK) and potentials (E^a and £ K ) for the voltage-gated sodium (Nav) and potassium (Kv) channels. Additionally, the membrane resistivity is replaced by a "leak" conductivity (GL) account­ing for the membrane resistivity along with other mecha­nisms related to the ion homeostasis (e.g., Na+ /Ca2 + and Na+ /K+ exchangers). See Hodgkin and Huxley (1952) and Koch (1999) for more details.

Both Cable Theory and Hodgkin-Huxley models can be described by one unique partial differential equation (PDE):

B-dv CV + D (7)

where V is the membrane potential, and A, B, C and D are model parameters provided in Table 1. In this table, £ i is chosen such that V = Viest at rest, i.e.,

EL = (1 +GNa/GL +GK /GL) Vrest

'm,a

(GNaENa + GKEK)/GL

(8)

Table 1 PDE model parameters

Parameter Cable theory Hodgkin-Huxley

Itd^

4ftT

B

C

( «my \ "

«0 i y ^ "my.» I Cmxd0 T ^ Cn¡yKd^o J d0 + ¿ - ¿ = 1

~JT ^rest my,0

rrry,0

Cm7tdp

ho

^ (GL + GNa + GK)

^ ( G L £ L +

GNa^Na + GK-EK)

Table 2 Nav and Kv channel Hodgkin-Huxley evolution equations (potential and time units are, respectively, mV and ms in this table)

Nav

GNa(V) = Gmm3h

^=am(V)(l-m)-fSm(V)m

f = ah(V)(l - h) - MV)h a (V) 25-(V-vaA)

m l ' lO^-C-f res tB/ lO- l )

ah(V) = 0.07e-(v-v«>A>/20

Kv

GK(V) = GKn4

| = « n ( V ) d -

a an w~ a n i v > ioo(edo-

^ ( V ) = 0.125e-

» ) - A i ( V ) «

"(V-Vrest) -(I'-VrestB/lO-l)

- (V-1W/80

Ph(V) 1 e ( 3 0 - ( y - y r e s t ) ) / i o + 1

In the special case of Hodgkin-Huxley, GN 3 and GK are not constant but depend on the time-dependent membrane potential V, and two constants C?Na and GK (Hodgkin and Huxley 1952; Koch 1999), see Table 2. In this table, the corresponding evolution equations describe the probability of open versus closed states of gating particles n, m and h. They need to be simultaneously opened in a given configu­ration (3 m's and 1 h for Nav; 4 m's for Kv) to allow for the full opening of the gate, see Hodgkin and Huxley (1952) and Koch (1999) for further information.

2.4 Damage coupling model

The lack of protective myelin of the NRs (Shreiber et al. 2009), recent findings highlighting the primordial role of blast-damaged ion channels in neurobehavioral deficits (Park et al. 2013), and the Nav leak observed during trauma (Wang et al. 2009) suggest the need for a strain-based damage cri­terion specific to the ion channels' properties. This Nav (and in this model, Kv) leak has been rationalized by a hyperpo-larizing shift (or left-shift) of transient current operational range (Wang et al. 2009) in the conductance of the ion chan­nel. Boucher et al. (2012) proposed to capture this stretch-induced alteration by left-shifting the potential for Nav chan­nels' activation and inactivation variables in the damaged part of the nodal axolemma. Despite the fact that such bleb-bing should a priori affect equally all types of ion channels,

previous attempts to consider the leak of Kv have not been fully pursued (Boucher et al. 2012). It has actually been sug­gested that the observed extracellular K+ accumulation in post-traumatic rat hippocampal slices would be due to glial activity and not neuronal leak (D'Ambrosio et al. 1999). In the absence of further body of evidence, damage linked to blebbing or other type of damage are equally considered for both Kv and Nav conductivity alterations.

The modeling approach of Boucher et al. (2012) involves the consideration of additional "left-shifted" m and h (see Table 2) calculated by increasing the potential V in am, «h, An and jSh by a value related to the degree of damage. These left shifted m and h are then weighed-averaged with their healthy counterparts to obtain the factor multiplying C?Na, see Table 2. We postulate here instead that both Nav and Kv are being affected by the microscopic membrane strain, and that their corresponding reversal potentials E^a and £ K should be tending toward 0 as the ion channels are increasingly leak­ing due to a loss of membrane integrity with increased dam­age (as dictated by the Nernst equation when ion concentra­tions on both sides of the membrane equilibrate themselves). Once a maximum value of microscopic membrane strain £ is reached, the ion channel reversal potentials remain at 0:

£Na(em)

£k(em) =

£Na,0 ( l

0, £ K , 0 (1 "

o,

<M.V\ if £ m < £

else if £ m < £

else

(9)

where £Na,o and £K,O are the reference potentials of Nav and Kv, and where y is an exponent representing the sensitivity of the damage model to small versus large deformation.

As can be seen in Table 2, the parameters am, an, ab, j3m, j3a and jSh are only functions of V. However, because these parameters are intrinsically related to the evolution of the ion channel potential, the accompanying parameters originally calibrated by Hodgkin and Huxley (1952) can a priori be considered to implicitly include the contribution of the corre­sponding reversal potentials ¿Na and £ K , SO that the parame­ters am, «h, An and jSh are actually functions of V - E^a, and aa and /3n, of V - E^, i.e., the difference between the mem­brane potential and the corresponding reversal potentials. As a consequence, an increase or decrease in the reversal poten­tials of Nav or Kv by a given value as proposed by Eq. (9) should be translated into a modification by the contrary of this value in the potentials of parameters am, an, a^, An, A and A- For a surface membrane strain £m, these parameters are thus given by:

am(V, £ m )

oth(V, £ m )

dm {V + (if)* E^o) , am (V + Emo) , ab(V+(f)r £Na,o); ah(V • -EWo)

if £m

else

if £ m

else < £

Pm(V,em)

Ph(V,em)

ttn(V,em)

Pn(V, £m)

\Pm(V+(fY £Na,o). 1 Pm {V + £Na,o) ,

' Ph(V+(^fY £Na,o): [Ph(V + £Na,o) ,

K^+W^o), an(V + £K,O) ,

ÍA.Ív + W f i K . o ) , I Pn (V + £K,o) ,

if em < £ else

if £ m < £

else

if £ m < £

else

if £ m < £

else (10)

Note that in the case of Nav, as ENa > 0, this approach is in agreement with the one proposed by Boucher et al. (2012) as damage will decrease ¿Na through Eq. (9) and thus increase V - £N a through Eq. (10).

Finally, these mechanical alterations automatically should a priori imply modifications of £L- Three possibilities were studied. The first choice assumed that the ion homeosta­sis exchangers would remain insensitive to all alterations and that £ L would thus remain constant. The second choice assumed that the ion homeostasis exchangers would not be damaged during deformation but would try to accommodate the changes in concentrations due to alterations of Nav and Kv by still following Eq. (8), where conductances and rever­sal potentials are modified following Eqs. (9) and (10). The third choice assumed the latter change with Eq. (8), but also an additional modification of £ i following a similar expres­sion as for ¿Na and £ K in Eq. (9). The first assumption led to strong oscillations insensitive to any input currents, a phe­nomenon which was not observed experimentally and was thus discarded. The third case naturally implies the additional identification of £ i -specific e and y (which are, a priori, dif­ferent from their Nav and Kv counterparts). Such identifica­tion is far from being straightforward as the signal was found to be weakly sensitive to £ i for small and large values of the microscopic strain, while switching abruptly to spontaneous firings in-between where the main triggering signals dies out quickly for moderate strain. As a consequence, the second assumption was chosen.

2.5 Finite difference discretization of the final model

Our in-house dedicated finite difference softwareNeurite was used for the discretization and calculations of Eq. (7). The approach consists of a ID finite difference scheme discretiz-ing the succession of myelinated IRs and NRs forming one cylindrical representative axon, see Fig. 1-3. Temporal and spatial convergences were systematically verified for all the simulations presented in this work.

The following relations are derived by considering the resistances, capacitances and conductances of unit deformed elements [characterized by Eqs. (5) and (6)] of given resis­tivities, capacities and conductivities.

For a given stretched spatial discretization step Ax m / N R = Axo(l + em,a )2 and an IR/NR-specific microscopic axial strain sm;a (superscripts are dropped subsequently for clar­ity), the axoplasm, membrane, myelin layer i and myelinated membrane element resistances are thus given for each region by

r _ 4pa(l+em,a) A ^

m,a jrdoAx

r l ~-' my

'"mm

where n

m,a (11)

^ m y . O ^

Z" m y j

t = l r n

my

my

0 for all NRs, and where use was made of Eqs. (5) and (6). Again, it must be emphasized that here and subsequently, Ax is the stretched discretization step.

Similarly, the individual membrane, myelin layer i and myelinated membrane element capacitances are given by

Lm —

é --umy

Cmm

CmJtdp

ftoyrr --Ax

m,a

Cmyjrdmyfi

y,oV1+e -Ax (12)

(l/Cm + Z S l / C m y ) " 1

Finally, the Hodgkin-Huxley leak and voltage dependent element conductances are given by

gL td0GL •Ax

gNa(V)

§K(V)

JTrfoGNa(y,£m) A T

M l + £ m , a )

jrdoGK(V,sm)

(13)

Ao(l+«m,a) -Ax

where G^a(V, em) and G^a(V, em) are the voltage and strain-dependent electrical conductivities of the Nav and Kv

channels. Finally, under the assumption that the finite num­ber of ion channels (albeit damaged) remains constant during deformation, the strain dependency of both electrical conduc­tances gNa and £K should only arise from the strain depen­dency of their conductivity counterparts GN 3 and GK- This is ensured by "normalizing" the membrane surface deformed area ndAx by its surface stretch 1 + em, making use of Eq. (5) and recalling that Ax = Axo(l + £m,a), thus leading to

the strain-independence of jrdpAx . The leak conductance Ao(l+Sm,a)'

gL is on the other hand left strain-dependent. This choice is motivated by the fact that the membrane resistivity in Eq.

2 In the Lagrangian framework followed by Neurite, Ax = Axo(l + e m a ) where Axo is the spatial discretization step for e m a = 0; Ax is then automatically refined by Neurite when needed, to ensure spatial convergence.

(11) is also strain-dependent, and that the activity ofthe ion homeostasis exchangers are assumed to be proportional to the membrane area as a first approximation.

3 Results

In the following, the analytical derivation of Eq. (3) is first analyzed in terms of irreversible damage evolution. A cali­bration is then proposed based on the experimental work of Shi and Whitebone (2006).

3.1 Damage analysis

where x~ = r]eq/E and r¡eq = (l/?jj~ + l/r]^) . In these expressions, e^ a is the microscopic axial strain at the onset of unloading, and ejE, is the corresponding accumulated damage strain. Two cases are then possible:

• The undamaged case (white region in Fig. 2):

'm,a

^ a

r+¿o / _imax \

I 1 - e é°T+ J 0

where r+ = r¡\ ¡E. • The damaged case (gray region in Fig. 2):

(15)

The post-loading strain relaxation equation proposed in the following is derived from the system of equations (3). For this section only, we assume axial axonal loadings at con­stant strain rate ¿o until a maximum applied strain emax is reached; the unloadings are then accomplished free of stress. For a given combination of strain and strain rate, two cases are thus possible: in the first one, the combination of strain and strain rate does not irreversibly damage the axon, whereas in the second one, damage is accumulated following the evo­lution equations below. Figure 2 shows a representation of both regions in a ¿o versus emax plot using the parameters calibrated in the next section.

Setting the time origin when emax is reached, i.e., at the onset of unloading, the analytical integration of the system of equations (3) yields

£m,a(í) = (e * m,a -D,a )e -t/T-

-D,a (14)

o.i

0.08

0.06

J 0.04

0.02 -

Fig. 2 Damage space ofthe analytical model: the vertical dashed line corresponds to ¿o and the red boundary line to the equation

-X+SQI n(l--^r-) where r^ TI + /E

£max ff

e^a = [E(l + a) - r*¿o] e ~ + (r*¿0 - aE)

'D,a l+o; v ^ a (4.a " £) (16)

where £ = OQ/E, a = E/k, x* = (1 + a)x+ and where

t* = -r+ln(l-——) (17) \ r+eoj

is the time at which damage first occurs when taking the onset of loading as the time reference.

Finally, in a configuration where the axon is clamped dur­ing stretch, the measurement of the output signal is made at a distance (1 + ea)L from the input signal, where L is the original axonal length. Because of the presence of tis­sue active molecular reorganization viscosity r]f, the axon is able to grow. As such, the ID discretized axon used for the simulation should be taken sufficiently long to account for this effect. The macroscopic axial strain ea(i) can be inferred from the microscopic one by noting that during the unload­ing:

KB m,a (18)

where K ii +i2

The mechanical model thus depends on five parameters: t+ (the time constant of the tensile axonal active molecu­lar reorganization), x~ (the equivalent time constant of the coupled stress-free axonal active molecular reorganization and viscosity of the surrounding structures), £ (the normal­ized damage threshold stress), a (a parameter characteriz­ing the membrane damage evolution), and K (the proportion of macroscopic over microscopic strain rates during relax­ation). It must be emphasized that these parameters can a priori be measured experimentally at the macroscopic tissue scale. The sixth and seventh parameters, y (the exponent rep­resenting the sensitivity ofthe damage model to small versus

large deformation), and e (the microscopic membrane strain threshold at which the ion channels lose their potential) are coupling them to the electrophysiological model, see Eqs. (9-10).

Unfortunately, experimental loadings such as the ones simulated in the following section cannot generally be per­formed at constant strain rates. In such cases, e^ a and ejí, are simply determined by numerical integration of the system of equations (3) for the chosen loading, i.e., here, by numer­ically incrementally applying the experimentally observed macroscopic strain and calculating both emA and eo at each increment. The resulting e ^ a andejí, are then used in Eq. (14) for the unloading (free of stress). Such numerical integration is done in the following section.

3.2 Calibration

The resulting model is Anally calibrated according to the experimental results of tensile tests of guinea pig spinal cord white matter of Shi and Whitebone (2006). It must first be emphasized that the model proposed here provides the elec­trophysiological signal propagation within one unique axon, i.e., one AP, whereas the experimental results used for cali­bration and validation involve CAPs from a section of spinal cord (Shi and Whitebone 2006). As such the approxima­tion is made that the simulated APs, deterministically deter­mined, can be calibrated against the experimental CAPs (see also Sect. 4 below). To this end, L «a 10 mm of myelinated guinea pig axon was discretized in our in-house dedicated finite difference software Neurite. The electrophysiological parameters for guinea pig spinal cord axons were taken from literature when possible, and estimated from other animal and/or organs when no data was available (see Supplemen­tary Material B).

Shi and Whitebone (2006) provide a measurement of the reduction in the CAP amplitude immediately after the tensile test and continuously for the subsequent 30 min. In these experiments, the white matter strip is subjected to a set of slow(¿o = 0.006-0.008 s_1) and fast maximum axial strain rates (¿o = 355-519 s_1) for mild (emax = 25 %), moderate (emax = 50 %) and severe (emax = 100 %) axial strains

loads. The authors recorded the exact evolution of the strain rate for each loading case, and the same loading was applied numerically to the set of equations (3) for all three cases.

Despite observations of stress concentrations at the NRs during stretch injury, the relation between axonal, myelin and glial stiffnesses, as well as the role of oligodendro­cytes and paranodal axo-glial adhesions in the "glueing" of all three components remain largely controversial (Shreiber et al. 2009). As a consequence and in the absence of more conclusive evidences, the case where all the strain is sus­tained equally by the NRs and the IRs is considered here (i.e., v = x). The methodology for the model calibration is provided in Supplementary Material B, with the final set of parameters in Tables 3, 4, and 5, and the final results for a triggering current of 0.04 nA during 3 ms are shown in Fig. 3. It is Anally important to emphasize the fact that effective val­ues of the resistivity and electric constant of the membrane or myelin layers are used here. The non-effective counterparts also used in other formulations can be recovered by divid­ing or multiplying the effective values by the thickness of the membrane or myelin layers. See Koch (1999) for more details. The new resting potential was evaluated by taking the minimum of the potential in the measured point for the last ms of each simulation and each AP was measured with respect to this value. A noticeable change in the resting potential was observed in the slow-severe and fast-mild cases. Addi­tionally, for these cases (and not in the other cases), new (one stable and the other one non-stable) equilibriums were observed at 30 min by which, once triggered by the input signal, the first output signal is followed by a train of other peaks. Subsequent signals are then much more attenuated that the first one (not shown). In the proposed work, only the first destabilizing peak was considered for the calibration, but further studies will be needed to classify such behavior, as it seems to indicate not only two (Volman and Ng 2013) but three regimes (stable-unstable-stable when increasing the damage).

A time step of 32.9868 ns (calculated as a factor 0.8 of the critical time step) and element sizes of 80 (im for the IRs and 2.1 (im for the NRs were used for the reference simula­tion. The deformed elements were then elongated according

Table 3 Geometrical model parameters Parameter

Membrane reference thickness (ho)

Axon reference diameter (do)

Number of myelin layers in the IRs (nmy)

Myelin layer reference thickness (/¡my,o)

Length of nodes of Ranvier

Internodal distance

Value

4nm

3 \im

=s45

^18nm

2.1 \im

^800 Lim

Reference

Koch (1999)

Ouyang et al. (2010)

Hildebrand and Hahn (1978)

Jensen et al. (2011)

Ouyang et al. (2010)

Jacobs and Cavanagh (1969), Ibrahim et al. (1995), Koch (1999)

Table 4 Electrical model parameters; note that effective values are used (non-effective values can be recovered by multiplication or division by the membrane/myelin layer thickness; see Koch 1999)

Parameter Value Reference

Axial cytoplasm resistivity (/oa)

Membrane resistivity (pm)

Myelin layer resistivity (pmy)

Membrane electric constant (Cm)

Myelin layer electric constant (Cmy)

Resting potential (Vrest)

Nav reversal potential (i'Na.o)

Kv reversal potential (£K,O)

Leak conductivity ( G L )

Nav reference conductivity (GN S )

Kv reference conductivity ( G K )

1.87 £2 m Koch (1999)

2 . 5 x l 0 9 i 2 m Koch (1999)

4.44 x 106 Í2 m Chomiak and Hu (2009)

4 x 10~ n F/m Hodgkin and Huxley (1952), Koch (1999), Chomiak and Hu (2009)

1.08 x 10-1 0 F/m Chomiak and Hu (2009)

-65 .5mV Boucher et al. (2012)

49.5 mV Hodgkin and Huxley (1952), Koch (1999)

-77 .5mV Hodgkin and Huxley (1952), Koch (1999)

1.2 x 10-8 S/m Hodgkin and Huxley (1952), Koch (1999)

4.8 x 10-6 S/m Hodgkin and Huxley (1952), Koch (1999)

1.44 x 10-6 S/m Hodgkin and Huxley (1952), Koch (1999)

Table 5 Calibrated mechanical model parameters

Parameter

r

a

K

Y

s

Value

18.08s

1.8 x 10-

0.9

111.5s

0.5

2

0.1

15 20 Time (min)

Fig. 3 CAP (experiment) and AP (simulation) amplitude evolution during unloading in percentage of the reference one (no strain): the experimental results from Shi and Whitebone (2006) {dashed lines), and our simulation results {solid lines) for slow and fast strain rate and for mild (25 %), moderate (50 %) and severe (100 %) maximum macro­scopic axial tensile strain

to the microscopic strain but did not need to be refined (con­vergence was still observed). The time step was recalculated accordingly.

It must be emphasized that the experimental work of Shi and Whitebone (2006) was used for calibration for its metic­ulous and complete analysis of CAP evolutions over a wide range of strain and strain rate. However, the model can easily be calibrated against any other system, providing that enough data points allow for the identification of all model parame­ters.

4 Discussion

4.1 Irreversible damage thresholds

A first observation is that the model naturally provides a set of thresholds for tissue/cell irreversible damage, see Fig. 2. Indeed, below a given threshold of oo/rft, a con­stant strain rate ¿o applied to the tissue is not overcoming its dynamic active accommodation (e.g., membrane/tissue turnover, (de)polymerization and (de)lipidation), and the membrane molecular bonds thus remain undamaged, inde­pendently of the applied strain. However, above this strain rate threshold, a strain threshold exists at

r^éolti (19)

above which irreversible microscopic damage eventually occurs. Any tensile test characterized by a (emax, ¿0) pair situated in the gray area of Fig. 2 will imply that a compo­nent of the AP amplitude decrease will never be recovered (as observed in all cases in Fig. 3). Accordingly, any pair situ­ated in the white area will guarantee full recovery of the elec­trophysiological functions after sufficient time has elapsed. Such behavior is in line with experimental results analyzing axonal growth under stretch on the one hand (Dennerll et al. 1989), and increased damage for increased strains and strain rates on the other hand (Shi and Whitebone 2006).

The proposed model also suggests that axons populated by cardiac Nav isotype (Nav 1.5), observed to provide a reversible behavior after stretch (Wang et al. 2009) would simply be characterized by a higher ao/r]+ factor and/or emax than the ones populated by the Nav 1.4 and Nav 1.6 "irreversible" isotypes, thus remaining in the non-damaged areaofFig. 2. TheNav 1.4andNav 1.6 isotypes, on the con­trary, would thus present a higher susceptibility to irreversible damage. The presence of the underlying cortical axoplasm and its intrinsic mechanical relation to the ion channel most likely conditions these parameters.

In addition to the electrophysiological data being routinely identified for all ion channels, we propose here the system­atical identification of the parameters of Eq. (19) for each one of the ion channels. This will eventually participate in categorizing not only the physiological properties of the ion channels, but also their propensión at sustaining or not irre­versible damage.

4.2 Model analysis

The results of Fig. 3 confirm the ability of the model to repro­duce the main features of the electrophysiological properties during unloading of a damaging injury. A discrepancy with the experimental results emerges for the slow case at moder­ate strain and for fast loading cases at moderate and severe tensile strains. For the slow case, a strong instability lead­ing to a sudden decrease in AP was observed when lowering further. Note that this sudden "switch" is actually happen­ing for the slow-severe and fast-mild cases and seems to be automatically accompanied by a change in the equilibrium potential to ~-55mV, consistent with the simulations of Boucher et al. (2012) for a left-shift of 20 mV. The simulation results are thus not incompatible with the experimental ones as the experimental CAPs are actually an average of multiple APs, for which an upper bound is simulated here. Future work on the averaging of simulated APs should tackle this issue. The fast case discrepancy is also not surprising considering the simple mechanical model proposed here: the overevalua-tion of the damage should be corrected with a more adequate extracellular matrix model. Current work is focusing on such extension.

As discussed earlier, the overall behavior of the model potential propagation might be driven by the leaking behavior of Nav channels, in agreement with experimental campaigns aimed at enhancing axonal conduction after SCIs by Nav

channel blockade (Hains et al. 2004). Our model exhibits two different alteration mechanisms. The first one is exclusively due to geometrical effects: the elongation of the axons at con­stant volume increases the membrane surface available to ion and electron exchanges with the extra-cellular medium, see Eq. (5). The incompressibility approximation is motivated by the assumption that minor swelling and fluid fluxes eventu­

ally leading to significant mass variations would only occur on larger time scales. This leads to a faster decrease in the potential, and eventually a sudden collapse of the signal if the signal reaching the Nav channels is not strong enough to acti­vate them. This mechanism alone was tested with the model proposed here and was shown to be insufficient to decrease the signals as much as observed experimentally by (Shi and Whitebone 2006). The second mechanism involves an addi­tional damage-based ion channel properties alteration, see Eqs. (9-10). It must be emphasized that this alteration solely depends on em and can thus be reversible or not depending on the mechanical behavior; see Eq. (19). Figure 4 shows the simulated potentials (a) and ion channel (first and twelfth NRs) currents (b), for (i) the reference case, and slow-severe case (at t = 30 min) with (ii) only with geometrical alteration (no ion channel damage alteration), (iii) without ion chan­nel damage alteration for Kv, and (iv) the full model. The effect of geometrical alteration on the potential is mainly measured in terms of delay of the signal (whether poten­tial in Fig. 4a or current in Fig. 4b); this is, however, not due to the stretch-induced elongation (as seen in the 12th NR of Fig. 4b, where damaged signal and reference sig­nal arrival times remain relatively close) but to the active growth of the axon (as seen in the arrival of the potential at the point of measurement). A video of the propagation of the potential versus microscopically deformed length is pro­vided as Supplementary Material C and shows that the dam­aged signal indeed travels at the same speed with respect to the microscopically deformed referential. However, the active growth of the axon due to the presence of tissue active molecular reorganization viscosity r]f implies a point of measurement in this referential at a further distance (both points of measurement are indicated by arrows in the video), thus confirming the delay of signal observed experimentally (Shi and Pryor 2002).

The model predicts a similar left-shift (see 1st NR of Fig. 4b) as reported by Wang et al. (2009) but does not seem to exhibit the immediate post-traumatic larger Nav currents reported in the same reference. However, the patch suction loading used in this reference implies a very localized dam­age mostly characterized by blebbing, i.e., strained mem­brane with underlying cortical axoplasm disruption. Figure 4 shows that such type of "leak" can be obtained by only accounting for geometrical alteration, i.e., without Eqs. (9-10). This could be explained by the fact that suction involves a very localized and severe membrane bending, possibly strain­ing and damaging the membrane without directly damaging the ion channels. Additionally, a large immediate increase in Nav current implies a priori a stronger potential and faster signal, which is in contradiction with most of the experi­mental campaigns making use of stretching (Shi and Pryor 2002; Shi and Whitebone 2006) or blast loading (Connell et al. 2011). However, the model exhibits different equilib-

Fig. 4 Simulated potentials (a) and ion channel (first and twelfth NRs) currents (b), for (i) the reference case, and slow-severe case (at t = 30 min) with (ii) the full model, (iii) only with geometrical alteration (no damage alteration), and (iv) without damage alteration for

(a)

>

c

-20

-40

-60

-80 30

(b) 200

u -200

30

40

40

' A' A ' ' ' '

1 1 1 A

J JJ \\

1

Reference Strain only

— No Kv damage

— Full model

1

-

A -

y v 50 60 70 80

Time (ms) 100 110 120

1 NR 12 NR Reference

Strain only No 1^ damage - -

Full model

50 60 70 80

Time (ms)

100 110 120

Table 6 Equilibrium ion channel currents

Case

Reference

Slow-mild

Slow-moderate

Slow-severe

Fast-mild

Fast-moderate

Fast-severe

Nav (pA)

-0.24

-0.51

-1.6

-5.52

-5.09

-7.91

-7.91

Kv (pA)

0.87

0.23

0.0012

-0.0019

-0.0024

0

0

rium current than in the reference case is observed for the damaged cases in Table 6, indicating that the damage-based leaks (if indeed mainly caused by this equilibrium currents alterations) are mostly affecting Nav. This observation sug­gests that the modeling choice of altering Kv along with Nav

shouldhave significant influence on the overall stability of the system but that the experimentally observed extracellular K+ longer term accumulation in post-traumatic rat hippocampal slices might indeed be due to glial activity and not neuronal leak (D'Ambrosio et al. 1999).

rium (i.e., without triggering signal) currents, see Table 6. These values are orders of magnitude lower than on Fig. 4 during AP propagation but the Nav equilibrium (with­out AP) current is drastically increased between the refer­ence cases and the damaged cases. This would indicate that Nav leaks observed post-trauma are either a slower mech­anism occurring at a later stage (Hains et al. 2004)—see also discussion below on damage time scales—or actually a more discrete phenomenon linked to a different equilib­rium involving a much larger ionic current without any AP propagation.

Finally, the simulations in Fig. 4 exhibited an important influence of the damage-based Kv alteration on the Nav cur­rent and AP. This influence is particularly marked in terms of stability of the AP and change in resting potential. Addi­tionally, the model is assuming that the parameters y and e in Eqs. (9) and (10) are the same for both Nav and Kv, thus indicating a need for a potential differentiated calibration of both sets of parameters. Additionally a much lower equilib-

4.3 Parameters analysis

It must be emphasized that other families of solutions can be found for other more realistic values of v accounting for the cross-links of the axon overlaps in the white matter. How­ever, the purpose of this article being mainly to introduce the model, the assumption of homogeneous deformation can be considered as a first satisfactory approximation for this tensile case. Additional experimental methods can actually participate in identifying the value of v. For instance, in the compression experiments of Ouyang et al. (2010), coher­ent anti-Stokes Raman spectroscopy and immunohistochem-istry identified that the ratio between the length and diam­eter of the NRs increased by a factor of four between rest and onset of unloading. Discarding a possible demyelination and assuming one more time the incompressibility of the NRs, the coefficient v can thus be estimated directly from the macroscopic strains for this loading condition. Further experimental work should systematically involve such mea-

surement to help identifying the value of v [note that para-nodal demyelination consideration coupled with appropriate ion channels density observation (Ichimura et al. 2005) could also be factored in in future models]. A complete parame­ter study is provided in Supplementary Material B for the remaining parameters.

Finally, it must be emphasized that the Hodgkin-Huxley model and its corresponding parameters are certainly not fully adequate for the guinea pig spinal cord. However, Fig. 3 exhibits a relatively good fit. Additionally, this work is aim­ing at providing a new framework for which mechanical and electrophysiological models can easily be changed for other more adequate models.

4.4 Membrane integrity

The use of a microscopic membrane strain em naturally defines a framework for a measure of membrane integrity. Shi and Whitebone (2006) assessed it by use of horseradish peroxidase (HRP) exclusion tests. In this reference, the den­sity of axons labeled with HRP (axons/mm2) was calculated for each case of Fig. 3 at í = 0 min and t = 30 min. Assuming that the microscopic membrane strain is directly proportional to the leak of HRP in one axon, we propose to corroborate the hypothesis that em can be used as a measure for membrane integrity by plotting the density of axons labeled with HRP (but divided by (1 +e a) to account for the volume-conserving cross-section reduction during stretch). The results are shown in Fig. 5. A linear tendency can be seen, confirming the abil­ity of em at capturing membrane integrity. Note that the two points at zero density were not taken into account for the lin-

20001 1 1 1 1 1 1 1

^ 5 ^ S •a "£ 1600-

g g 1200-a o /

i—i q _ ,

•si <D

a -a t = Slow moderate Slow severe Fast mild Fast moderate Linear interpolation

= 0 min •

• • •

t = 30 min

& 0

0 Zl

O ^ ^ r O ' ' ' ' ' ' 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Em

Fig. 5 Density of axons labeled with HRP per cross-section deformed area (Shi and Whitebone 2006) versus corresponding model micro­scopic membrane strain em, and linear regression; the points corre­sponding to zero density were discarded from the linear regression as a membrane disruption smaller than the size of HRP (5 nm) cannot be detected (Shi and Whitebone 2006)

ear regression as the membrane holes that allow the passage of HRP are estimated to be ~5nm large, which implies that disruption smaller than the size of HRP will go undetected by HRP (Shi and Whitebone 2006).

4.5 Damage time scales

A similar study focused on long-term (up to 14 days) electro­physiological dysfunctions after blast trauma has highlighted the need for the consideration of another time scale (Park et al. 2013). More specifically, whereas the model proposed here accounts for two alteration mechanisms—namely, geo­metrical deformation and immediate ion channel damage— this experimental work highlights the role of neurofilament expression alteration. After an injury, neurofilaments have been observed to accumulate in the soma, suggesting an alter­ation of the transport mechanisms in axons and thus changes in axonal caliber (Park et al. 2013). As demonstrated by the model proposed here, geometrical deformation of the axon invariably leads to changes in the signal propagation. Such expression alteration will then further decrease the ability of the axon to propagate an electrical signal on a longer-term basis (days).

Additionally, axonal injury has been shown to lead to cleavage of all-spectrin (a cortical axoplasm protein), either attacked by Ca2+ proteases, originally overactivated by Na+ /Ca2 + exchangers (Wang et al. 2009) overwhelmed by a possible initial Nav leak, or simply mechanically sheared (Park et al. 2013). This cleavage then reduces the associa­tion of this protein with ankyrin G (a protein involved in the anchoring of Nav channels), thus leading to a diffusion and reorganization of Nav channels at the NRs (Park et al. 2013). Again, such change is suggested to lead to a decrease in AP on a long-term basis.

In addition to the geometrical and damage alterations highlighted above, both observations point toward a third alteration mechanism due to longer-term mechanisms such as diffusion (Nav in the membrane) or active transport alter­ation (neurofilaments in the axoplasm). The model pro­posed here is built on the same code as the one pro­posed by Garcia et al. (2012) for the simulation of axonal growth based on diffusion and active transport of proteins. Future work could focus on the coupling of the diffu­sive and active transport mechanisms of the latter model with the one proposed here. This should then allow for the consideration of the longer-term mechanisms discussed here.

Other possible mechanisms could involve a slower time-dependent recalibration of £ L through Eq. (8) as the ion homeostasis exchangers try to reaccommodate the changes in potentials due to Nav and Kv after damage to rebalance the potential to its resting potential.

4.6 Future work

In addition to the longer-term mechanism discussed above, the natural extensions of this model are pointing toward a three-dimensionalization of the framework. This will involve the adoption of less restrictive solvers such as the finite ele­ment method. Such extension is needed for the identifica­tion of more complex loading states such as local compres­sion, blast, inertia, or more generally loadings involving a stress gradient within the thickness of the axon. Even though the model proposed here should a priori behave appropri­ately under large macroscopic deformation because only the microscopic deformation (a fraction of the former) is con­cerned by the small strain elastic law, a large deformation framework should enhance the proposed formulation con­siderably and will be proposed along with the finite element formulation.

The observation of strong instabilities for different level of damage and different parameters (especially taking into account, or not, Kv alterations), as well as the fact that once a new resting potential has been reached, subsequent APs seem to behave differently (in general much more attenu­ated) will be tackled in future work. This includes aspects of AP propagation, such as a deeper study of unstable salta­tory behaviors in damaged myelinated axons (Boucher et al. 2012) or in demyelinated axon (Waxman and Brill 1978; Coggan et al. 2010) and their relation to mechanical loading.

Finally, as highlighted earlier, the model proposed here provides the AP propagation within one unique axon. The experimental results used for calibration and validation involve CAPs. As such, a more genuine comparison should be done by averaging (based on the corresponding experi­mental methods used for the measurement) the behavior of many axons using random statistics related to their diame­ters, the exact positions of the NRs or the lengths of the IRs. Future work will tackle this effort.

5 Conclusion

We propose here a new multiscale framework coupling mechanical and electrophysiological properties. To the best of the knowledge of the authors and as suggested by a recent review (Gupta and Przekwas 2013), this model is the first of the kind. It is constituted of four main building blocks whose parameters have been taken from actual experiments, leaving only seven remaining free parameters. These parameters are not only representative of purely mechanical properties— measurable experimentally at the macroscopic level—but also of the molecular reorganization or damage that would be expected in living tissue under traumatic stress. The model was eventually thoroughly utilized to analyze recent works on ion channel leak and membrane integrity, among others,

and the different time scales possibly involved in the cor­responding mechanical damage were discussed. As a con­clusion, this model paves the road to a new series of models able to predict the alteration of neuronal electrophysiological functions under external damaging load (both reversible and irreversible), thus linking mechanical injuries to subsequent acute functional deficits. By directly linking macroscale mechanical loading, microscale deformation, molecular reor­ganization and functional deficits, such model ultimately par­ticipates to the overarching goal of understanding the causes and consequences of spinal cord and traumatic brain injuries.

Acknowledgments The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP7 2007-2013)/ERC Grant Agreement No. 306587, from the Spanish Ministry of Science (TIN2010-21289-C02-02), and the Cajal Blue Brain Project, the Span­ish partner of the Blue Brain Project. The authors gratefully acknowl­edge the computer resources, technical expertise and assistance pro­vided by the Supercomputing and Visualization Center of Madrid (CeSViMa).

References

Alford P, Dabiri B, Goss J, Hemphill M, Brigham M, Parker K (2011) Blast-induced phenotypic switching in cerebral vasospasm. Proc Natl Acad Sci 108:12705-12710

Babbs CF, Shi R (2013) Subtle paranodal injury slows impulse con­duction in a mathematical model of myelinated axons. PLoS One 8(7):e67767

Boucher P-A, Joós B, Morris CE (2012) Coupled left-shift of Na„ chan­nels: modeling the Na+-loading and dysfunctional excitability of damaged axons. J Comput Neurosci 33:301-319

Cernak I, Noble-Haeusslein L (2010) Traumatic brain injury: an overview of pathobiology with emphasis on military populations. J Cereb Blood Flow Metab 30:255-266

Chomiak T, Hu B (2009) What is the optimal value of the g-Ratio for myelinated fiber in the rate CNS ? A theoretical approach. PLoS ONE 4:e7754

Cloots RJH (2011) Multi-scale mechanics of traumatic brain injury. PhD thesis, Eindhoven University of Technology

Cloots RJH, van Dommelen JAW, Kleiven S, Geers MGD (2013) Multi-scale mechanics of traumatic brain injury: predicting axonal strains from head loads. Biomech Model Mechanobiol 12:137-150

ConnellS, Gao J, Chen J, ShiR (2011) Novel model to investigate blast injury in the central nervous system. J Neurotrauma 28:1229-1236

Coggan JS, Prescott SA, Bartol TM, Sejnowski TJ (2010) Imbalance of ionic conductances contributes to diverse symptoms of demyeli-nation. Proc Natl Acad Sci 107(48):20602-20609

D'Ambrosio R, Maris DO, Grady MS, Winn HR, Janigro D (1999) Impaired K+ homeostasis and altered electrophysiological proper­ties of post-traumatic hippocampal gha. J Neurosci 19:8152-8162

Dennerll TJ, Lamoureux P, Buxbaum RE, Heidemann SR (1989) The cytomechanics of axonal elongation and retraction. J Cell Biol 109:3073-3083

García JA, Peña JM, McHugh S, Jerusalem A (2012) A model of the spa­tially dependent mechanical properties of the axon during its growth. Comput Model Eng Sci 87:411-432

Goldstein L, Fisher A, Tagge C, Zhang X-L, Velisek L, Sullivan J, Upreti C, Kracht J, Ericsson M, Wojnarowicz M, Goletiani C, Maglakelidze G, Casey N, Moncaster J, Minaeva O, Moir R, Nowinski C, Stern

R, Cantu R, Geiling J, Blusztajn J, Wolozin B, Ikezu T, Stein T, Budson A, Kowall N, Chargin D, Sharon A, Saman S, Hall G, Moss W, Cleveland R, Tanzi E, Stanton P, McKee A (2012) Chronic trau­matic encephalopathy in blast-exposed military veterans and a blast neurotrauma mouse model. Sci Transí Med 4:1-16

Gupta RK, Przekwas A (2013) Mathematical models of blast-induced TBI: current status, challenges, and prospects. Front Neurol 4(59):1-21

Hains BC, Saab CY, Lo AC, Waxman SG (2004) Sodium channel blockade with phenytoin protects spinal cord axons, enhances axonal conduction, and improves functional motor recovery after contusion SCI. Exp Neurol 188:365-377

Hicks R, Fertig S, Desrocher R, Koroshetz W, Pancrazio J (2010) Neu­rological effects of blast injury. J TRAUMA Inj Infect Crit Care 68:1257-1263

HildebrandC,HahnR (1978) Relation between myelin sheath thickness and axon size in spinal cord white matter of some vertebrate species. J Neurol Sci 38:421-434

Hodgkin A, Huxley A (1952) A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol 117:500-544

Ichimura H, Shiga T, Abe I, Hara Y, Terui N, Tsujino A, Ochiai N (2005) Distribution of sodium channels during nerve elongation in rat peripheral nerve. J Orthop Sci 10:214-220

Ibrahim M, Butt A, Berry M (1995) Relationship between myelin sheath diameter and internodal length in axons of the interior medullary velum of the adult rat. J Neurol Sci 133:119-127

Jacobs M, Cavanagh J (1969) Species differences in internode formation following two types of peripheral nerve injury. J Anat 105:295-306

Jensen T, Bech M, Bunk O, Menzel A, Bouchet A, Due GL, Feiden-hans'l R, Pfeiffer F (2011) Molecular X-ray computed tomography of myelin in a rat brain. Neurolmage 57:124-129

Jerusalem A, Dao M (2012) Continuum modeling of neuronal cell under blast loading. Acta Biomaterialia 8:3360-3371

Koch C (1999) Biophysics of computation. Oxford University Press, Oxford

Markvin VS, Tanelian DL, Jersild RA Jr, Ochs S (1999) Biomechanics of stretch-induced beading. Biophys J 76:2852-2860

Moore D, Jerusalem A, Nyein M, Noels L, Jaffee M, Radovitzky R (2009) Computational biology—modeling of primary blast effects on the central nervous system. Neurolmage 47:T10-T20

Nyein M, Jason A, Yu L, Pita C, Joannopoulos J, Moore D, Radovitzky R (2010) In silico investigation of intracranial blast mitigation with relevance to military traumatic brain injury. Proc Natl Acad Sci 107:20703-20708

Ochs S, Jersild RA Jr, Pourmand R, Potter CG (1994) The beaded form of myelinated nerve fibers. Neuroscience 61:361-372

Ouyang H, Galle B, Li J, Nauman E, Shi R (2008) Biomechanics of spinal cord injury: a multimodal investigation using ex vivo guinea pig spinal cord white matter. J Neurotrauma 25:19-29

Ouyang H, Sun W, Fu Y, Li J, Cheng J, Nauman E, Shi R (2010) Compression induces acute demyelination and Potassium channel exposure in spinal cord. J Neurotrauma 27:1109-1120

Park E, Eisen R, Kinio A, Baker AJ (2013) Electrophysiological white matter dysfunction and association with neurobehavioral deficits fol­lowing low-level primary bast trauma. Neurobiol Dis 52:150-159

Peter S, Mofrad M (2012) Computational modeling of axonal micro­tubule bundles under tension. Biophys J 102:749-757

Shi R, Pryor JD (2002) Pathological changes of isolated spinal cord axons in response to mechanical stretch. Neuroscience 110:765-777

Shi R, Whitebone J (2006) Conduction deficits and membrane disrup­tion of spinal cord axons as a function of magnitude and rate of strain. J Neurophysiol 95:3384-3390

Shreiber D, Hao H, Elias R (2009) Probing the influence of myelin and glia on the tensile properties of the spinal cord. Biomech Model Mechanobiol 8:311-321

Volman V, Ng LJ (2013) Computer modeling of mild axonal injury: implications for axonal signal transmission. Neural Comput 25:1— 36

Wang JA, Lin W, Morris T, Banderali U, Juranka PF, Morris CE (2009) Membrane trauma and Na+ leak from Na„ 1.6 channels. Am J Phys­iol Cell Physiol 297:C823-C834

Waxman SG, Brill MH (1978) Conduction through demyelinated plaques in multiple sclerosis: computer simulations of facilitation by short internodes. J Neurol Neurosurg Psychiatry 41:408-416

Wu L, Tjahjanto D, Becker G, Makradi A, Jerusalem A, Noels L (2013) A micro-meso-model of intra-laminar fracture in fiber-reinforced composites based on a discontinuous Galerkin/cohesive zone method. Eng Fract Mech 104:162-183